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 \date\today
 
 \begin{abstract}
+In this paper we follow up the study of `complex-complex landscapes'  \cite{one}, rugged landscapes in spaces of many complex variables.
+Contrary to the real case, the index of saddles is not itself relevant, as it is always half the total dimension. The relevant topological objects here are Stokes trajectories, gradient lines joining two saddles, where Lefschetz thimbles merge. The well-studied `threshold level', separating regions with minima from regions with saddles in the real case, here separates regions where Stokes lines are rare, from region where they proliferate.
+Likewise, when a real landscape is prolonged to complex variables, the distinction between "one step replica-symmetry breaking" and "many step replica symmetry breaking"  is that in the former case the saddles are at first free of Stokes lines, while in the latter these immediately proliferate.
 \end{abstract}
 
 \maketitle